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Hodge theory is a beautiful synthesis of geometry, topology, and anal- ysis, which has been developed in the setting of Riemannian manifolds. On the other hand, spaces of images, which are important in the math- ematical foundations of vision and pattern recognition, do not fit this framework. This motivates us to develop a version of Hodge theory on metric spaces with a probability measure. We believe that this constitutes a step towards understanding the geometry of vision. We extend the theory of the Laplacian on domains of Euclidean space or on a manifold and next to metric space. Finally we give a beautiful application in web researcher engin.
 

This set is an introduction to a method for studying Constant Mean Curvature Surfaces (CMC) and Minimal Surfaces with harmonic maps . In recent years , integrable systems approaches for the construction of CMC surfaces in R3 have been developed . One of them , using finite type solutions of the sinh-Gordon equation .Another approach to CMC surfaces was developed by J. Dorfmeister, F. Pedit, and H. Wu in 2006 . It is commonly called the DPW method. The Gauss map of a CMC surface is a harmonic map , and in the DPW method every such harmonic map can be obtained from some holomorphic potential in a loop algebra . This method gives a description of all immersed CMC surfaces in R3 . in this paper... 

Morse 2-functions are higher-dimensional analogs to morse functions. A morse 2-function is a mapping to 2-dimensional disk with its critical points satisfying certain generality conditions. The goal of defining morse 2-functions and trisections is to generalize methods of 3-dimensional manifolds to dimension 4. A morse function on a 3-dimensional manifold leads to a decomposition of that manifold to two manifolds with boundary with common boundary. This decomposition is called Heegaard splitting. This decomposition is unique up to stabilization. Trisections are 4-dimensional analogs of Heegaard splittings and decompose manifold into three parts. The intersection of each pair is a solid genus... 

Causal discovery is of utmost importance for agents who must plan and decide based on observations. Since mistaking correlation with causation might lead to un- wanted consequences. The gold standard to discover causal relation is to perform experiments. However, experiments are in many cases expensive, unethical or impossible to perform. In these situations, there is a need for observational causal discovery. Causal discovery in the observational data setting involves making significant assumptions on the data and on the underlying causal model. This thesis aims to alleviate some of the assumptions and tries to identify the causal relationships and causal mechanisms using generative neural... 

The Yau-Tian-Donaldson conjecture is a challenging problem in complex geometry which currently caught the attention of some birational geometers. This conjecture is about a formulation of the problem of finding the best metric over a Kahler manifold in terms of an asymptotic GIT stability condition. The first stability condition was proposed by Mumford in ICM 1962 which surprisingly just needs the vector bundle to be“more ample” among its subbundles. Another important theorem by Narasimhan and seshadri provides a connection between the slope stability and the existence constant curvature connections for the flat holomorphic vector bundles over compact Rieman surfaces. In 1983, Donaldson... 

In this paper, the field theory tools will be used to study none-equilibrium statistical processes and eventually analyze the dynamic of the neocortex. Assuming the neocortex is Markovian, a model is proposed which contains fluctuations of the neuron activities as well as response to the stimulations. The experimental data shows that the fluctuation and correlation have a vital impact on the dynamics of the cortex, and many of its characteristics can only be justified by it.The paper will study the model that considers correlations and fluctuations to reach a more accurate evaluation of the cortex than the mean field approximation  

The manifolds that will play the central role in this thesis are b-Poisson manifolds. These manifolds are a special sub-class of Poisson manifolds which are in many ways close to being symplectic: For π the Poisson structure dual to a symplectic form, the top wedge π^nnever meets the zero section of Λ^2n TM (non-degeneracy of the symplectic form). We define a b-Poisson manifold to be a Poisson manifold (M,π)such that π^n vanishes transversally to the zero section in Λ^2n TM. A pivotal result about the dynamics of integrable systems is the Liouville-Arnold-Mineur theorem (or action-angle coordinate theorem), which states that the compact common level sets of the integrals f_iare tori that... 

We intend to find conditions that lead to a posterior probability distribution becoming convex in terms of $-\log$. To achieve this, we will leverage modern optimal transport tools and their geometric results. Then, in relevant examples related to the theory of Bayesian mean field approximation, we aim to enhance convex models, enabling more precise and improved results. We do this task in the case of Gaussian mixture models. Also, we introduce a new definition of convexity on discrete spaces and strive to utilize this definition for the aforementioned estimation. Additionally, the presented definition can be studied independently  

In this thesis the elastic energy of plates is computed. Primarily, some physical related concepts such as stress, strain and strain energy density are defined. Then based on rigidity estimate theorem, the gradient map on bounded Lipschitz domains is approximated by a constant rotation map R.Thanks to this approximation theorem,we prove the precompactness of sequences of finite bending energy, and the limiting energy of two dimensional thin plates is shown to be an elastic energy. This limit innovation is called Γ –convergence introduced in 1993. It will indeed be revealed that the induced elastic energy of a deformable body is the limit of the sequence of stored energy of body under... 

Let (M; g) be a Riemannian manifold. Using classical Stokes’ theorem one can show that the equality (dω; η)L2 = (ω; δη)L2 holds for smooth forms ! and η with compact supports, where δ is the formal adjoint of d . There are some examples of Riemannian manifolds for which the above equality does not hold for general forms ! and η i:e: smooth square-integrable forms such taht d! and δη are also squareintegrable. In the case that the above equality holds for such general forms on a Riemannian manifold (M; g) , we say that the L2 - Stokes theorem holds for (M; g) . In 1952, Gaffney showed that the L2 - Stokes theorem holds for complete Riemannian manifolds. But at that time, there was no powerful... 

One of the important questions in mathematics is generalized. In case of metric spaces, one of the questions is: Can we extend notions of Riemannian geometry to arbitrary metric spaces smoothly? Ricci curvature is one of the most important concepts in differential geometry. Ricci curvature is defined for the Riemannian manifold and has many applications in mathematics and physics like Einstein’s equation in relativity theory. There is a good notion for a metric space having “sectional curvature bounded below by K” or “sectional curvature bounded above by K”, due to Alexandrov. Can we define the concept of Ricci curvature in metric space? A motivation for this question comes from Gromov’s... 

As the core of this thesis, we express the relation between optimal transportation and convex geometry especially the variational approach to solve Alexandrov problem, which leads to a geometric interpretation to adversarial models and propose a novel framework for these models. Along the way, we peruse generative adversarial networks from optimal transportation view and show that generator calculates the transportation map and the discriminator computes Wasserestein distance, which is equivalent to Kantorovich potential. By using optimal mass transportation theory and choosing an especial cost function c, we see that the generator and discriminator are equivalent. Therefore, once the... 

We introduce a refinement of the Ozsváth-Szabó complex associated by Juhász [8] to a balanced sutured manifold (X; ). An algebra A is associated to the boundary of a sutured manifold. For a fixed class s of a Spinc structure over the manifold X, which is obtained from X by filling out the sutures, the Ozsváth-Szabó chain complex CF(X; ; s) is then defined as a chain complex with coefficients in A and filtered by the relative Spinc classes in Spinc(X; ). The filtered chain homotopy type of this chain complex is an invariant of (X; ) and the Spinc class s 2 Spinc(X). The construction generalizes the construction of Juhász. It plays the role of CF (X; s) when X is a closed three-manifold,...